No need to leave. Even if OLCI fails just to to a GA and ask they check you in.
If need checked bags, JFK agents can through check across tickets. Just ask. DL will also protect across tickets in cases of IRROPS.
DL operates from T2, T3, and TBIT (aka TB).
JFK flights should arrive T2 normally. Hawaii flights depend where inbound equipment is coming from. Although IME, they're likely from TBIT. There's airside bus between T2/3 and TBIT.